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On the ample vector bundles over curves in positive characteristic. (English) Zbl 1062.14042
Here the authors prove the following extension to positive characteristic of a result of D. Barlet, T. Peternell and M. Schneider [Math. Ann. 286, 13–25 (1990; Zbl 0716.32011)]. Let \(E\) be an ample vector bundle over a smooth projective curve defined over an algebraically closed field of positive characteristic. The authors construct a family of curves in the total space of \(E\), parametrized by an affine space, that surjects onto the total space of \(E\) and gives a deformation of a (nonreduced) zero section of \(E\). The main tool is a strong theorem concerning the Harder-Narasimhan filtration of the Frobenius pull-backs of \(E\) proved by A. Langer [Ann. Math. (2) 159, 251–276 (2004; Zbl 1080.14014)].

MSC:
14H60 Vector bundles on curves and their moduli
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[1] Barlet, D.; Peternell, T.; Schneider, M., On two conjectures of Hartshorne’s, Math. ann., 286, 13-25, (1990) · Zbl 0716.32011
[2] Hartshorne, R., Ample subvarieties of algebraic varieties, Lecture notes in math., vol. 156, (1970), Springer-Verlag Berlin · Zbl 0208.48901
[3] Langer, A., Semistable sheaves in positive characteristic, Ann. math., 159, 251-276, (2004) · Zbl 1080.14014
[4] Ramanan, S.; Ramanathan, A., Some remarks on the instability flag, TĂ´hoku math. J., 36, 269-291, (1984) · Zbl 0567.14027
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